We develop a new technique to prove the conditional CLT for the weighted bootstrap mean. Through 0-1 laws, we show that this conditional CLT can be derived from an unconditional one which easily arises (conditioning with respect to the weights) from the standard Lindeberg CLT.
"A zero-one law approach to the central limit theorem for the weighted bootstrap mean." Ann. Probab. 24 (1) 532 - 540, January 1996. https://doi.org/10.1214/aop/1042644731